CahnHilliardtype

Cahn-Hilliard type refers to a class of partial differential equations that model phase separation in binary mixtures and related systems, characterized by a gradient-flow structure that preserves the overall amount of material. The equations describe the evolution of an order parameter u(x,t) representing composition or concentration, which tends to separate into distinct phases while conserving mass.

The canonical Cahn-Hilliard equation has the form ∂u/∂t = ∇·(M(u) ∇μ), where μ is the chemical potential given by μ = δF/δu = f'(u) − κΔu. Here F[u] = ∫ (f(u) + (κ/2) |∇u|^2) dx is an energy functional, f(u) is a double-well potential favoring two pure phases, κ > 0 is a gradient-energy coefficient, and M(u) is a mobility. Many Cahn-Hilliard type models generalize this structure by allowing variable mobility M(u), nonlocal free-energy terms, higher-order or coupled dynamics, or stochastic forcing.

Key properties of CH-type models include mass conservation, energy dissipation (the free energy F[u] decreases over time), and the appearance of phase-separated patterns that coarsen as time progresses. Solutions exhibit spinodal decomposition and often form complex microstructures depending on boundary conditions and parameters. Boundary conditions are typically no-flux, preserving total mass within the domain.

Applications span materials science, polymer physics, metallurgy, and electrochemical modeling, where CH-type equations are used to study microstructure evolution and stability. Numerical treatment emphasizes energy-stable schemes, handling of stiffness, and adaptive methods to capture interfaces, with common variants incorporating fluid flow or nonlocal interactions.